Phase Transitions in Quantum Triangular Ising antiferromagnets

Ying Jiang

Inst. Theor. Phys., Univ. Fribourg, Switzerland

Y.J. & Thorsten Emig, PRL 94, 110604 (2005)Y.J. & Thorsten Emig, PRB 73, 104452 (2006)

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Introduction

Non-frustrated Ising system: LiHoF4

[Ronnow et al, Science 308, 389 (2005); Bitko et al, PRL 77, 940 (1996)]

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Triangular Ising Antiferromagnets (TIAF)

Geometrical frustration

Highly degenerated ground states: exactly one frustrated bond per triangle

T = 0Extensive entropy density

Spin correlation: algebraic decay

[Wannier, Hautappel (1950)]

?

Classical antiferromagnetic Ising system

[Stephenson (1970)]

Macroscopic degeneracy Continuous symmetry of the system

For triangular Ising antiferromagnets

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Triangular Ising Antiferromagnets (TIAF)

Quantum system

Quantum fluctuation order from disorder ?

/J

T/J

QLRO QCPOrder ?

dis

ord

erPhase diagram ?

Transverse field: intends to flip spins

Zero exchange field flippable spins

T = 0

Quantum critical point expected

T ≠ 0

Competition between thermal and quantum fluctuations

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Spin--string mapping in classical 2D TIAF

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From spin configuration to dimer covering

Hardcore dimer covering on dual lattice

Properties of classical TIAF ground states

Height profile

on sites of lattice

:dimer crossed

:no dimer crossed

single spin flip:

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From dimer covering to fluctuating lines

+

Dimer covering Reference pattern Fluctuating lines

++++

+++++

+++-

-- -

- -

-

--

--

---

---

+ ++++

++

++

+- -

--

---

--

-

+++

+++ +

+++ -

-

-

---

---

- -+ ++

++++

+-

---

--

---

-

+ +++

+++

++-

- --

--

--- -+ ++ ++ --- -

non-zero entropy density fluctuation

reference covering directed

geometrical frustration non-crossing

frustrated Ising spin configuration fluctuating strings

XOR

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Free energy functional of strings

Lock-in potential

average string distance

displacement field

Global offset of flat strings

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The lock-in potential

Lock-in potential

irrelevant

quantum fluctuations increase the string stiffness

relevant

Equivalent flat states: shifts by a/2

2D self-avoiding non-crossing strings = 1D free fermions

stiffness

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Spin—spin correlations

++++

+++++

+++-

-- -

- -

-

--

--

---

---

+ ++++

++

++

+- -

--

---

--

-

+++

+++ +

+++ -

-

-

---

---

- -+ ++

++++

+-

---

--

---

-

+ +++

+++

++-

- --

--

--- -+ ++ ++ --- -

Spin-spin correlation

Vortex pair

stiffness

system unstable with defects

T=0 no defect quasi-long range ordered phaseT≠0 unbound defects disordered phase

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Phase diagram of quantum TIAF

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From 2D quantum system to classical 3D system

mapping to 3D classical system (Suzuki-Trotter theorem)

2D Quantum system

correspondence becomes exact

size in imaginary time direction

T=0: real 3D system T≠0: finite size 3D system

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Mapping to stacked string layers

spin-height relation

Spin-string mapping

3D XY model + 6-clock term

Topological defects

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Universality class of quantum phase transition

Decoupling of layers?

No!

QCP: 3d XY Universality

[Korshunov, (1990)]

p-fold clock term is irrelevant attransition point for 3D if

[Aharony, Birgeneau, Brock and Litster, (1986)]

Hs = 3D XY Hamiltonian + 6-fold clock term

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Quantum critical point

Decoupling of “spin waves” + topological defects (Villain mapping)

Villain coupling

Dimensional crossover approach for layered XY models[Ambegaokar, Halperin,Nelson and Siggia, 1980]

[Schneider and Schmidt, 1992]

~ 2/3 (3D XY)

Quantum phase transition point

Simulation: c/J ~ [Isakov & Moessner, 2003]Renormalization effects of clock term increases

}

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Phase boundaries

Finite size scaling approach[Ambegaokar, Halperin, Nelson & Siggia (1980); Schneider and Schmidt, 1992]

Relevance of the 6-clock term

[José, Kadanoff, Kirkpatrick and Nelson (1977)]

Phase boundaries at

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(C)

(PM)

(O)

Log-rough strings with bound defects

Strings locked-in by clock term

Phase diagram of quantum TIAF

[Monte Carlo Simulations, Isakov & Moessner, 2003]

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Summary

Transverse field TIAF system stacked 2D string lattice

Strongly anisotropic 3D XY model with 6-clock term obtained in a microscopic way

Quantum critical point 3D XY universality

Reentrance of the phase diagram due to the frustration and thecompetition between the thermal and quantum fluctuations

Phase diagram in excellent agreement with the recent simulations